Variable neighborhood search approaches for scheduling jobs on parallel machines with sequence-dependent setup times, precedence constraints, and ready times

In this paper, we discuss a scheduling problem for jobs on identical parallel machines. Ready times of the jobs, precedence constraints, and sequence-dependent setup times are considered. We are interested in minimizing the performance measure total weighted tardiness that is important for achieving good on-time delivery performance. Scheduling problems of this type appear as subproblems in decomposition approaches for large scale job shops with automated transport of the jobs as, for example, in semiconductor manufacturing. We suggest several variants of variable neighborhood search (VNS) schemes for this scheduling problem and compare their performance with the performance of a list based scheduling approach based on the Apparent Tardiness Cost with Setups and Ready Times (ATCSR) dispatching rule. Based on extensive computational experiments with randomly generated test instances we are able to show that the VNS approach clearly outperforms heuristics based on the ATCSR dispatching rule in many situations with respect to solution quality. When using the schedule obtained by ATCSR as an initial solution for VNS, then the entire scheme is also fast and can be used as a subproblem solution procedure for complex job shop decomposition approaches.     

Scheduling jobs on parallel machines with setup times and ready times

In this research we are interested in scheduling jobs with ready times on identical parallel machines with sequence dependent setups. Our objective is to minimize the total weighted tardiness. As this problem is NP-Hard, we develop a heuristic to solve this problem in reasonable time. Our approach is an extension of the apparent tardiness cost with setups (ATCS) approach by [Lee, Y. H., Pinedo, M. (1997). Scheduling jobs on parallel machines with sequence dependent setup times. European Journal of Operational Research, 100, 464–474.] to allow non-ready jobs to be scheduled – meaning we allow a machine to remain idle for a high priority job arriving at a later time. To determine the scaling parameters for our composite dispatching rule (called ATCSR), we first develop a ‘grid approach’ that considers multiple values for the scaling parameters, generates multiple schedules, and chooses the best schedule for the solution. This experimentation was then used to develop regression equations to predict the values of the scaling parameters that would yield the highest quality solution. The grid and regression versions of ATCSR provide better performance than grid and empirically based formula versions of ATCS, BATCS, and X-RM which are the prominent algorithms in the literature.     

Scheduling unrelated parallel batch processing machines with non-identical job sizes and unequal ready times

This research analyzes the problem of scheduling a set of n jobs with arbitrary job sizes and non-zero ready times on a set of m unrelated parallel batch processing machines so as to minimize the makespan. Unrelated parallel machine is a generalization of the identical parallel processing machines and is closer to real-world production systems. Each machine can accommodate and process several jobs simultaneously as a batch as long as the machine capacity is not exceeded. The batch processing time and the batch ready time are respectively equal to the largest processing time and the largest ready time among all the jobs in the batch. Motivated by the computational complexity and the practical relevance of the problem, we present several heuristics based on first-fit and best-fit earliest job ready time rules. We also present a mixed integer programming model for the problem and a lower bound to evaluate the quality of the heuristics. The small computational effort of deterministic heuristics, which is valuable in some practical applications, is also one of the reasons that motivates this study. The results show that the heuristic proposed in this paper has a superior performance compared to the heuristics based on ideas proposed in the literature.     

Scheduling reentrant jobs on parallel machines with a remote server

This paper explores a specific combinatorial problem relating to reentrant jobs on parallel primary machines, with a remote server machine. A middle operation is required by each job on the server before it returns to its primary processing machine. The problem, which is new to the literature, is inspired by the logistics of a semi-automated microbiology laboratory. We establish the NP-hard nature of the problem, and demonstrate various structural properties. A heuristic is developed and tested on randomly generated benchmark data. No alternative heuristics are available in the literature for comparison, but results indicate solutions reliably within 1.5% of optimum. Moreover, tests of our heuristic on real-life data from the microbiology laboratory provide a 20% improvement in throughput relative to current practice.     

Online batch scheduling on parallel machines with delivery times

We study the online batch scheduling problem on parallel machines with delivery times. Online algorithms are designed on m parallel batch machines to minimize the time by which all jobs have been delivered. When all jobs have identical processing times, we provide the optimal online algorithms for both bounded and unbounded versions of this problem. For the general case of processing time on unbounded batch machines, an online algorithm with a competitive ratio of 2 is given when the number of machines m=2 or m=3, respectively. When m≥4, we present an online algorithm with a competitive ratio of 1.5+o(1).     

Integrating non-preemptive open shops scheduling with sequence-dependent setup times using advanced metaheuristics

In this article, we consider non-preemptive open shops scheduling problem (OSSP) where setup times are sequence-dependent (SDST) on each machine to minimize makespan. The contributions of this article are threefold. Firstly, we incorporate a very practical assumption in our problem, SDST, which, according to Allahverdi et al. (2008) [Allahverdi, A., Ng, C. T., Cheung, T. C. E., & Kovalyov, Y. M. (2008). A survey of scheduling problems with setup times or costs. European Journal of Operational Research, 187(3), 985–1032], no paper has ever attempted to integrate into OSSP. Secondly, we propose two new advanced metaheuristics: multi-neighborhood search simulated annealing and hybrid simulated annealing to tackle the problem at hand. Thirdly, for the first time, we adapt two well-known constructive heuristics: longest total processing time and longest total remaining processing from the literature so as to consider the case of SDSTs. We also apply genetic algorithm from the literature of OSSP to embrace the concepts of SDST. Since there is no standard SDST-OSSP benchmark, we make certain adaptations on the Taillard’s benchmark [Taillard, E. (1993). Benchmarks for basic scheduling problems. European Journal of Operational Research, 64, 278–285] to include setup times. An experimental design based on foregoing benchmark is conducted to evaluate the competitiveness and robustness of our proposed algorithm against some effective algorithms in the literature. The obtained results strongly support the high performance of our proposed algorithms with respect to other well-known heuristic and metaheuristic algorithms.     

Scheduling independent jobs on uniform parallel machines to minimize tardiness criteria

The problem of scheduling N jobs on M uniform parallel machines is studied. The objective is to minimize the mean tardiness or the weighted sum of tardiness with weights based on jobs, on periods or both. For the mean tardiness criteria in the preemptive case, this problem is NP-hard but good solutions can be calculated with a transportation problem algorithm. In the nonpreemptive case the problem is therefore NP-hard, except for the cases with equal job processing times or with job due dates equal to job processing times. No dominant heuristic is known in the general nonpreemptive case. The author has developed a heuristic to solve the nonpreemptive scheduling problem with unrelated job processing times. Initially, the algorithm calculates a basic solution. Next, it considers the interchanges of job subsets to equal processing time sum interchanging resources (i.e. a machine for a given period). This paper models the scheduling problem. It presents the heuristic and its result quality, solving 576 problems for 18 problem sizes. An application of school timetable scheduling illustrates the use of this heuristic.     

Scheduling unrelated parallel batch processing machines with non-identical job sizes

Scheduling unrelated parallel batch processing machines to minimize makespan is studied in this paper. Jobs with non-identical sizes are scheduled on batch processing machines that can process several jobs as a batch as long as the machine capacity is not violated. Several heuristics based on best fit longest processing time (BFLPT) in two groups are proposed to solve the problem. A lower bound is also proved to evaluate the quality of the heuristics. Computational experiments were undertaken. These showed that J_SC-BFLPT, considering both load balance of machines and job processing times, was robust and outperformed other heuristics for most of the problem categories.     

Scheduling unrelated parallel machines with optional machines and jobs selection

In this paper we study two generalizations of the well known unrelated parallel machines scheduling problem under makespan (C_max) minimization. First, a situation in which not every available parallel machine should be used and it is desirable to employ only a subset of the parallel machines. This is referred to as “Not All Machines” or NAM in short. This environment applies frequently in production shops where capacity exceeds demand or when production capacity can be lent to third companies. Also, NAM can be used to increase production capacity and it is not clear how many additional machines should be acquired. The second studied generalization has been referred to as “Not All Jobs” or NAJ. Here, there is no obligation to process all available jobs. We propose Mixed Integer Programming mathematical formulations for both NAM and NAJ, and it is shown that the latter can be effectively solved with modern commercial solvers. We also present three algorithms to solve the NAM problem. These algorithms are compared with the proposed MIP formulation when solved with IBM ILOG CPLEX 12.1. Comprehensive computational and statistical experiments prove that our proposed algorithms significantly improve the results given by the solver.     

Scheduling jobs on identical machines with agreement graph

We consider the following problem of scheduling with agreements: a set of jobs must be scheduled non-preemptively on identical machines subject to constraints that only some specific jobs can be scheduled concurrently on different machines. These constraints are represented by an agreement graph and the aim is to minimize the makespan. This problem is NP-hard. We study the complexity of the problem for two machines and arbitrary bipartite agreement graphs, in particular we prove the NP-hardness of the open problem proposed in the literature which is the case of two machines with processing times at most 3. We propose list algorithms with empirical results for the problem in the general case.     

The unbounded single machine parallel batch scheduling problem with family jobs and release dates to minimize makespan

In this paper we consider the unbounded single machine parallel batch scheduling problem with family jobs and release dates to minimize makespan. We show that this problem is strongly NP-hard, and give an O(n(n/m+1)^m) time dynamic programming algorithm and an O(mk^k+1P^2k−1) time dynamic programming algorithm, where n is the number of jobs, m is the number of families, k is the number of distinct release dates and P is the sum of the processing times of all families. We further give a heuristic with a performance ratio 2. We also give a polynomial-time approximation scheme for the problem.     

Scheduling jobs on a single batch processing machine with incompatible job families and weighted number of tardy jobs objective

In this paper, we minimize the weighted and unweighted number of tardy jobs on a single batch processing machine with incompatible job families. We propose two different mixed integer linear programming (MILP) formulations based on positional variables. The second formulation does not contain a big-M coefficient. Two iterative schemes are discussed that are able to provide tighter linear programming bounds by reducing the number of positional variables. Furthermore, we also suggest a random key genetic algorithm (RKGA) to solve this scheduling problem. Results of computational experiments are shown. The second MILP formulation is more efficient with respect to lower bounds, while the first formulation provides better upper bounds. The iterative scheme is effective for the weighted case. The RKGA is able to find high-quality solutions in a reasonable amount of time.     

Scheduling on parallel machines with preemption and transportation delays

This paper deals with an identical parallel machines scheduling problem, where independent jobs can be preempted and transported from one machine to another. The transportation of a preempted job requires a time called the transportation delay. The goal is to find a solution that minimizes the total completion time (makespan). We first study the case of equal-size jobs where new complexity results are given. Then, to solve the problem with two identical machines, we present a dynamic programming algorithm and a fully polynomial time approximation scheme (FPTAS). Experimental results show the efficiency of the FPTAS compared to a previously published heuristic.     

A hybrid genetic heuristic for scheduling parallel batch processing machines with arbitrary job sizes

This paper investigates the scheduling problem of parallel identical batch processing machines in which each machine can process a group of jobs simultaneously as a batch. Each job is characterized by its size and processing time. The processing time of a batch is given by the longest processing time among all jobs in the batch. Based on developing heuristic approaches, we proposed a hybrid genetic heuristic (HGH) to minimize makespan objective. To verify the performance of our algorithm, comparisons are made through using a simulated annealing (SA) approach addressed in the literature as a comparator algorithm. Computational experiments reveal that affording the knowledge of problem through using heuristic procedures, gives HGH the ability of finding optimal or near optimal solutions in a reasonable time.     

Heuristic scheduling of jobs on parallel batch machines with incompatible job families and unequal ready times

This research is motivated by a scheduling problem found in the diffusion and oxidation areas of semiconductor wafer fabrication, where the machines can be modeled as parallel batch processors. We attempt to minimize total weighted tardiness on parallel batch machines with incompatible job families and unequal ready times of the jobs. Given that the problem is NP-hard, we propose two different decomposition approaches. The first approach forms fixed batches, then assigns these batches to the machines using a genetic algorithm (GA), and finally sequences the batches on individual machines. The second approach first assigns jobs to machines using a GA, then forms batches on each machine for the jobs assigned to it, and finally sequences these batches. Dispatching and scheduling rules are used for the batching phase and the sequencing phase of the two approaches. In addition, as part of the second decomposition approach, we develop variations of a time window heuristic based on a decision theory approach for forming and sequencing the batches on a single machine.     

Scheduling identical parallel machines and operators within a period based changing mode

This article addresses the problem of scheduling n non-preemptive jobs on m identical parallel machines with ω operators to minimize the makespan. The number of operators is less than the number of machines. As a result, it may happen that an operator has to supervise simultaneously several machines. This in turn has an impact on the job processing times as they become a function of the operators modus operandi. In this paper we consider the case where each modus operandi can only be changed at the end of a given period, i.e., a period based changing mode. We present heuristic algorithms to solve this problem. An experimental study is conducted to evaluate the performance of these heuristics in the average case, and a worst case analysis is presented for one of them.     

Scheduling machine-dependent jobs to minimize lateness on machines with identical speed under availability constraints

In this paper we consider the problem of scheduling n$n$ preemptive jobs on m$m$ machines with identical speed under machine availability and eligibility constraints when minimizing maximum lateness (_Lmax$(L_{\max }$). The lateness of a job is defined to be its completion time minus its due date, and _Lmax$L_{\max }$ is the maximum value of lateness among all jobs. We assume that each machine is not continuously available at all time throughout the planning horizon and each job is only allowed to be processed on specific machines. Network flow technique is used to formulate this scheduling problem into a series of maximum flow problems. We propose a polynomial time two-phase binary search algorithm to verify the feasibility of the problem and to solve the scheduling problem optimally if a feasible schedule exists. Finally, we show that the time complexity of the algorithm is O((n+(2n+2x))³log(UB-LB))$O((n+(2n+2x){)}^3\log (\text{<italic>UB</italic>}-\text{<italic>LB</italic>}))$. Most literature in parallel machine scheduling assume that all machines are continuously available for processing and all jobs can be processed at any available machine throughout the planning horizon. But both assumptions might not be true in some practical environment, such as machine preventive maintenance and machines that have different capabilities to process jobs. This type of scheduling problem is seldom studied in the literature. The purpose of this paper is to examine the scheduling problem with machines with identical speed under machine availability and eligibility constraints. The objective is to minimize maximum lateness. We formulate this scheduling problem into a series of maximum flow problems with different values of _Lmax$L_{\max }$. A polynomial time two-phase binary search algorithm is proposed to verify the feasibility of the problem and to determine the optimal _Lmax$L_{\max }$.     

Exact algorithms for a scheduling problem with unrelated parallel machines and sequence and machine-dependent setup times

A scheduling problem with unrelated parallel machines, sequence and machine-dependent setup times, due dates and weighted jobs is considered in this work. A branch-and-bound algorithm (B&B) is developed and a solution provided by the metaheuristic GRASP is used as an upper bound. We also propose a set of instances for this type of problem. The results are compared to the solutions provided by two mixed integer programming models (MIP) with the solver CPLEX 9.0. We carry out computational experiments and the algorithm performs extremely well on instances with up to 30 jobs.     

A branch-and-price algorithm for the general case of scheduling parallel machines with a single server

We consider the strongly NP-hard problem of scheduling two-operation non-preemptable jobs on two identical parallel machines. A single server, that can handle at most one job at a time, is available to carry out the first (or setup) operation. The second operation, to be carried out on the same machine but without the server, must be executed immediately after the setup. The objective is to minimize the makespan. We apply a column generation method to a population of partial schedules, in turn generated by some well known heuristics, to achieve effective and efficient solutions. We compare the performance of this method with those proposed earlier and also suggest future work.